Question with proof involving conjugacy classes

Prove that if a group G has an odd order than no other than is conjugate to its inverse.

I'm thinking that since divides , both and are odd since . If I assume that there is conjugate to other than 1, does it then follow that either or is even? How can I find or so I can get a contradiction?

I think i'm going in the right direction but i'm not sure. Any help would be appreciated.

Prove that if a group G has an odd order than no other than is conjugate to its inverse.

I'm thinking that since divides , both and are odd since . If I assume that there is conjugate to other than 1, does it then follow that either or is even? How can I find or so I can get a contradiction?

I think i'm going in the right direction but i'm not sure. Any help would be appreciated.

suppose then for any natural number we have let then is odd because |G| is odd. hence which gives us